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Chapter 1 Electromagnetic Field Theory

1-1 Electric Fields and Electric Dipoles

Gauss’s law of E : ∫∫∫ ∫∫ ==⋅

→

'1 dvQS d E s

ρ ε ε

∫∫∫ ∫∫∫ =⋅∇ → →

''

1'

vv

theormdivergence

dvdv E ρ ε ε

ρ =⋅∇⇒

→

E and ε0=910

36

1 −×π

( F /m) in

the free space.

For q at→

' R , field point at→

R 2

'

|'|4

→→

→

−=⇒

R R

aq E RR

πε 3|'|4

)'(→→

→→

−

−=

R R

R Rq

πε and

'

'

'

R R

R Ra RR

−

−= .

→

E d!e to a s"ste# of discrete char$es% ∑=

→→

→→→

−

−=

n

k k

k k

R R

R Rq E

1 3|'|

)'(

4

1

πε

&ol!#e so!rce ρ⇒ ∫∫∫ ∫∫∫ →

→∧→

==' ' 3

2'

||4

1'

4

1v v

R dv

R

Rdv

Ra E

ρ

πε

ρ

πε

'!rface so!rce ρ s⇒ ∫∫ ∧→

=' 2

'4

1 s

s R dS R

a E ρ

πε ine so!rce ρl ⇒

∫ ∧→

=' 2

'4

1

l

l R dl R

a E ρ

πε

Eg. how that Coulom!’s law 3

21

2

21

44 R

Rqq

R

qqa F R

πε πε ==

∧→" where

R

Ra R

= " R R = .

(roof)∵ →

= E q F 2

and E r

qS d E 21 4π

ε

==⋅ →→

∫∫ ,

3

1

2

1

44 R

Rq

R

qa E R

πε πε

== ∧→

∴ 2

21

4 R

qqa F R

πε

∧→=

Eg. Determine the electric field intensity of an infinitely long line charge of a

uniform density ρl in air.

('ol.)

∫∫ ∫ ∫ ==⋅→→

s

L

rLE dz Erd S d E 0

2

02

π

π φ

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0

2ε

ρ π

LrLE l = ,

r a E l r

02πε

ρ ∧→

=

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Eg. Determine the electric field intensity of an infinite planar charge with a

uniform surface charge density ρs.

('ol.) ∫ ==⋅→→

s EA ES S d E 22 ,

0

2ε

ρ A EA s=

<−=

>=

=⇒∧

∧

→

0,2

0,2

0

0

z z

z z

E s

s

ε

ρ

ε

ρ

Eg. # line charge of uniform density ρl in free space forms a semicircle of radius

b. Determine the magnitude and direction of the electric field intensity at the

center of the semicircle. $高考%

('ol.) φ πε

φ ρ sin

4

)(2

0

⋅−=→

b

bd E d l

y ,b

yd b

y E y E l l y

00

0 2sin

4 πε

ρ φ φ

πε

ρ π ∧∧∧→−=−== ∫

Eg. Determine the electric field caused !y spherical cloud of electrons with a

&olume charge density ρ'- ρ( for b R ≤≤0 )!oth ρ( and b are positi&e* and ρ'(

for R+b. $交大電子物理所%

('ol.)

(a) b R ≥

3

03

4bQ

π ρ −= , 2

0

3

0

2

0 34

R

ba

R

Qa E R R

ε

ρ

πε

∧→

−==

(*) b R ≤≤0

E a E R

∧→= , dS aS d R

∧→= , ∫ ∫ ==⋅

→→

i i s s R E dS E S d E 24π

∫∫∫ ∫∫∫ −=−==v v

RdvdvQ 3

00

3

4π ρ ρ ρ ,

0

0

3ε

ρ Ra E R

∧→

−=

Eg. # total charge Q is put on a thin spherical shell of radius b. Determine the

electrical field intensity at an ar!itrary point inside the shell. $台大電研%

('ol.)

24 b

Q s

π ρ = ,

−=

2

2

2

2

1

1

04 r

dS

r

dS dE s

πε

ρ

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α α coscos2

2

2

2

1

1

r

dS

r

dS d ==Ω , 0

coscos40

=

Ω

−Ω

=α α πε

ρ d d dE s

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Electric dipole: + pair of e!al *!t opposite char$es -ith separation.

2311

4

)2

()2

(|2

|

2

32/3

2

3

2/32

22/33

Rd R R

Rd R R

d d R R

d R

d R

d R

→→

−−

→→

−

−−

→→

→→

−

→→

⋅+≅⋅−≅

+⋅−=−⋅−=−

2

31|

2|

2

33

R

d R R

d R

→→

−−

→→ ⋅−≅+

−

⋅=

−

⋅≅

+

+−

−

−=⇒

→→→→

→→→→

→→

→→

→→

→→

→

P R R

P R

Rd R

R

d R

R

q

d R

d R

d R

d R

q E

2323333

4

13

4

2

2

2

2

4 πε πε πε

=⋅

−==

→→∧∧∧→θ θ θ θ cossincos Raa z R

+=⇒

∧∧→

θ θ πε

θ sincos24 3

aa R

E R

Eg. #t what &alue of θ does the electric field intensity of a z -directed dipole ha&e

no z -component.

('ol.) )sincos2(4 3

0

θ θ πε

θ

∧∧→

→

+= aa R E r , θ θ θ sincos ∧∧∧ −= aa z r

o z co#ponent 0sinsincoscos2 =⋅−⋅⇒ θ θ θ θ 2tan2 =⇒ θ ⇒! =4. or 12.3

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1-, tatic Electric otentials

→→→

∫ ⋅−=−⇒−∇= l d E " " " E

2

112

and ε ρ −=∇ " 2

Electric potential due to a point charge:

R

qdRa

R

qa"

R

R Rπε πε 44 2

=⋅−= ∫ ∞∧∧

)11

(4 12

21 12 R R

q" " "

−=−=πε

Electric potential due to discrete charges:

∑= −

=n

k k

k

R R

q"

1 '4

1

πε

Electric potential due to an electric dipole:

)11

(4 −+

−= R R

q"

πε

5f d 66 R, -e ha7e

+≅

−≅ −

−

+

θ θ cos2

1cos2

1 1

1

R

d R

d R

R

and

−≅

+≅ −

−

−

θ θ cos2

1cos2

1 1

1

R

d R

d R

R

→→

== d q Rqd " ,4

cos2πε θ )(

4 2

" Ra " R

πε

∧→

⋅=⇒

)sincos2(4

3 θ θ

πε θ θ θ

∧∧∧∧→

+=∂∂

−∂∂

−=−∇=⇒ aa R

" a

R

" a" E R R

calar electric potential due to &arious charge distri!utions:

&ol!#e so!rce ρ⇒ ∫∫∫ ='

'4

1

vdv

R"

ρ

πε .

'!rface so!rce ρ s⇒ ∫∫ ='

'41

s

s dS R

" ρ πε

ine so!rce ρl ⇒ ∫ = '4

1dl

R" l ρ

πε

ote: 1. " is a scalar, *!t→

E is a 7ector.

2. " E −∇=→

is 7alid onl" in the static 89 field.

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Eg. /!tain a formula for electrical field intensity along the a0is of a uniform line

charge of length L. The uniform line-charge density is ρl . $高考%

('ol.)2

,' L z z z R >−=

∫ − −=

2/

2/0

'

'

4

L

L

l

z z

dz "

πε

ρ ( )( ) 2

,2/

2/ln

4 0

L z

L z

L z l >

−+

=πε

ρ

( )[ ] 2,

2/4 22

0

L z

L z

L z

dz

d" z E l >

−=−=

∧∧→

πε

ρ

Eg. # finite line charge of length L carrying uniform line charge density ρl is

coincident with the x -a0is. Determine V and E in the plane !isecting the line

charge.

('ol.)

−

++

=

+= ∫ − y

L y

L

y #

d#" l

L

L

l ln22

ln24

2

2

0

2/

2/ 22

0πε

ρ

πε

ρ

and " E −∇=→

( )

+=

∧

220 2/

2/

2 y L

L

y y l

πε

ρ

Eg. # charge is distri!uted uniformly o&er an L× L suare plate. Determine V

and E at a point on the a0is perpendicular to the plate and through its center.

('ol.)2 L

Q s = ρ , 222 z y y +→ , ∫ −

+−

+++

=

2/

2/

2222

2

0

ln22

ln2

L

L

s dy z y L

z y L

" πε

ρ

+

⋅−

−+

++

= −

2

2

2

1

2

2

2

2

2

0

22

2tan

222

222

ln2

z L

z

L

z

L z

L

L z

L

L

L

Q

πε

+

=−∇= −∧→

2

2

2

1

2

0

22

2tan

z L

z

L

L

Q z " E πε

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Eg. # positi&e point charge Q is at the center of a spherical conducting shell of an

inner radius Ri and an outer radius Ro. Determine→

E and V as functions of the

radial distance R. $高考

%

('ol.) R: Ro,0

24

ε π Q R E S d E

s==⋅∫∫

→

, 2

04 R

Q E

πε = , ∫ ∞ =−=

R

R

Q EdR"

04πε

Ri6 R6 Ro, 0= E ,o R

Q

R R" "

00 4πε =

==

R6 Ri, 2

04 R

Q E

πε = , $

R

Q$ EdR" +=+−= ∫

04πε

−+=⇒

−= ioio R R R

Q

" R R

Q

$

111

4

11

4 00 πε πε

Eg. # charge Q is distri!uted uniformly o&er the wall of a circular tu!e of radius

b and height h. Determine→

E and V on its a0is )a* at a point outside the tu!e" )!*

at a point inside the tu!e.

('ol.) ( ) ( )∫ −+

=

−+

= π

ε

ρ

πε

φ ρ 2

0 22

0

22

0

'2

'

'4

''

z z b

bdz

z z b

dz bd d" s s

(a)( ) ( ) ( ) 22

22

00 22

0

ln2'2

'

h z bh z

z bbb

z z b

bdz " s

h s

o

−++−

++=

−+= ∫ ε

ρ

ε

ρ ,

bh

Q s

π ρ

2=

( )

+−

−+=−=

∧∧→

22220

11

2 z bh z b

b z

dz

d" z E s

oε

ρ

(*) ( ) ( )∫ ∫ −++−+=

h

z

s z

s

i z z b

bdz

z z b

bdz

" 22

0

0 22

0 '2

'

'2

'

ε

ρ

ε

ρ

( ) ( ) ( )

−++−⋅++= 2222

2

0

ln2

z hb z h z b z b

b s s ρ

ε

ρ

7/23/2019 new EM1


Eg. Consider two spherical conductors with radii b1 and b, )b,+b1* that are

connected wire. The distance !etween the conductors is &ery large in comparison

to b, so that charges on spherical conductors may !e considered uniformly

distri!uted. # total charge Q is deposited on the spheres. Find )a* the charges onthe two spheres" and )!* the electric field intensities at the sphere surfaces2

('ol.)

(a)20

2

10

1

44 b

Q

b

Q

πε πε = ,

2

1

2

1

b

b

Q

Q= , QQQ =+ 21

;21

2

2

21

1

1bb

bQand Q

bb

bQ

+=

+=

(*) 2

20

2

22

10

1

1

44

8

b

Q E and

b

Qnn

πε πε

== ,1

2

2

1

2

1

2

2

1

b

b

Q

Q

b

b

E

E =

=

Eg. /!tain a formula for the electric field intensity on the a0is of a circular dis3

of radius b that carries uniform surface charge density ρs. $高考%

('ol.) '''' φ d dr r ds = , 22 'r z R +=

( )∫ ∫ +=

π

φ πε

ρ 2

0 0 2/1220

'''

'

4

b s d dr

r z

r " ( )[ ] z b z s −+=

2/122

02ε

ρ

z " z " E

∂

∂−=−∇=

∧→

( )[ ]

( )[ ]

<++−

>+−

= −∧

−∧

0,12

0,1

22/122

0

2/122

0

zb z z z

zb z z z

s

s

ε

ρ ε

ρ

+s z ::1 ( ) 2

22/122

21 z

bb z z −≅+⇒

−,

( )2

0

2

4 z

b z E s

πε

ρ π ∧→

=

<−

>

=∧

∧

04

04

2

0

2

0

z z

Q z

z z

Q z

πε

πε

Eg. 4a3e a two-dimensional s3etch of the euipotential lines and the electricfield lines for an electric dipole.

('ol.)

For an electric dipole, 2

04

cos

R

qd "

πε

θ = =constant⇒ θ cosvc R =

E k l d = , -here k is a constant.

++=++

∧∧∧∧∧∧

φ φ θ θ φ θ φ θ θ E a E a E ak d Ra Rd adRa R R R sin

φ θ

φ θ θ

E

d R

E

Rd

E

dR

R

sin

== ,θ θ

θ sincos2 Rd dR = , θ 2sin E c R =

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1-5 4agnetic Fields

4agnetic field: % & &

'

−==0 µ µ

, -here (0=4<10 ( A/m) in the free space.

4agnetic flu0 density: ∫ ×

=$

R

R

ad ) & 2

0

4

π

µ

#mpere’s law of ' : ) l ' =⋅∫ d ⇔ * ' =×∇

% * * * & m

o

×∇+=+=×∇

µ

1 or * %

& =−×∇ )(

0 µ ,

) l d ' % &

' =⋅⇒−= ∫

0 µ , and ' ' % ' & r m µ µ χ µ µ 000 )1()( =+=+= ,

-here 01 µ

µ

χ µ =+= mr

Gauss’s law of & : ∫∫ =⋅⇔=⋅∇S

S & & 0d0

∵ 0=⋅∇ & , A∃ f!lfills A & ×∇=

A A A * &2

)( ∇−⋅∇∇=×∇×∇==×∇ µ

>hoose * A A µ −=∇⇒=⋅∇ 20 (ote: ε

ρ −=∇ " 2

is scalar oisson?s e!ation)

∵ '

4

1

'

dv

R

" "

∫∫∫ =

ρ

πε ,∴ '

4 '

dv

R

* A

" ∫∫∫ =

π

µ (+b/m)

4agnetic flu0: ∫∫ ∫ ∫∫ ⋅=⋅×∇=⋅=ΦS $ S

l d AS d AS d & ')( (+b)

6iot-a&art’s law: ∫ ∫ ×

=×

='

32

'

4

'

4$

R

R

Rl d )

R

al d ) &

π

µ

π

µ

∫ ∫∫∫ ==''

4d7'

4$ "

R

l d )

R

* A

π

µ

π

µ , ∵ , - , - , - ×∇+×∇=×∇ )()( ),

∴

∫ ∫ ∫ ∫ ×=

×∇+×∇=×∇=

×∇=×∇=

'

2

''

'4

')1('14

)'(4

'4

$

R

$ $ R

al d ) dl R

dl R

) R

l d ) R

l d ) A &π

µ π

µ π

µ π

µ

ote:

=×∇⇒++=

∂

∂+

∂

∂+

∂

∂=∇

0'''''

l d dz zdy yd# #l d

z z

y y

# #

, and then

32

'

4

)'

(

4 R

Rl d )

R

al d ) &d R ×

=×

=

π

µ

π

µ

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Eg. # direct current I flows in a straight wire of length , L. Find the magnetic

flu0 density & at a point located at a distance r from the wire in the !isecting

plane.

('ol.) ')'('' rdz a z z r adz z Rl d r φ =−×=× ,

21

22 )( r z R +=

∫ − +=

+=

L

L

oo

r Lr

)La

r z

rdz ) a &

2223222

)'(

'

4

π

µ

π

µ φ φ

Eg. Find the magnetic flu0 density at the center of a suare loop" with side w

carrying a direct current I .

('ol.)2

. L = ,

2

.r = in this case,

.

) z

...

. )

z & o

o

π

µ

π

µ 22

)2

()2

(2

2

2422

=+

×=

Eg. Find the magnetic flu0 density at a point on the a0is of a circular loop of

radius b that a direct current I .

('ol.) '' φ φ bd al d = , ba z z R r −= ,

2122)( b z R +=

'@'' 2 φ φ d bbzd a Rl d r +=×

2322

22

0 2322

2

)(2'

)(

4 b z

)b z d

b z

b z bz a ) &

or o

+=

+

+= ∫

µ φ

π

µ π

. 5n case of z =0,

b

) z & o

2 µ

=

Eg. Determine the magnetic flu0 density at a point on the a0is of a solenoid with

radius b and length L" and with a current in its N turns of closely wound coil.

('ol.) = &d ')(

.)'(2

23

22

2

0 dz L

/

b z z

)b z

+−

µ

[ ]

+−

−−

+=

++

+−

−== ∫ 22222221220

)(2)(2 b L z

L z

b z

z

L

/)

b z

z

b z L

z L

L

/) &d & oo

L µ µ

7/23/2019 new EM1


#mpere’s law of & : ∫ =⋅⇔=×∇c

) l d & * µ µ A , -here (= (0 in the free space.

Eg. #n infinitely long" straight conductor with a circular cross section of radius b

carries a steady current I . Determine the magnetic flu0 density !oth inside andoutside the conductor. $交大光電所%

('ol.)

(a) 5nside the cond!ctor, br ≤ %

∫ ∫ ====⋅1

2

2

22

0)()(2d

$

oo ) b

r )

b

r r& &rd l & µ

π

π µ π φ

π

=: 22A

b

r) a

o

π

µ φ =

(*) B!tside the cond!ctor%r

) a & ) r&l d &

o

$

o

π

µ µ π φ

22

2

=⇒==⋅∫ Eg. # long line carrying a current I folds !ac3 with semicircular !end of radius

b. Determine magnetic flu0 density at the center point P of the !end. $高考%

('ol.)

21 & & & += , -hereb

) z & o

π

µ

421 ⋅=

,

b

) z & o

4

2

µ =

Eg. # current I flows in the inner conductor of an infinitely long coa0ial line and

returns &ia the outer conductor. The radius of the inner conductor is a" and the

inner and outer radii of the outer conductor are b and c" respecti&ely. Find the

magnetic flu0 density & for all regions and plot & &ersus r . $高考電機技師%

('ol.) ar ≤≤0 , 22

a

r) a &

π

µ φ = , br a ≤≤ ,

r

) a &

π

µ φ

2=

cr b ≤≤ ,r

)

bc

r ca &

π

µ φ

2)(

22

22

−

−=

Eg. Determine the magnetic flu0 density inside an infinitely long solenoid with

air core ha&ing n closely wound turns per unit length and carrying a current I .

('ol.) nL) &L o µ = =: n) & o µ =

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Eg. The figure shows an infinitely long solenoid with air core ha&ing n closely

wound turns per unit length. The windings are slanted at an angle

α and carry a

current I . Determine the magnetic flu0 density !oth inside and outside the

solenoid.

('ol.)

>=

<<=

br r

bn) a

r

bn) a

br

&,

sin

2

sin2

0,0

001 α µ

π

α πµ φ ϕ

,

>

<<=

br

br n) z &

o

,0

0,cos

2

α µ ,

21 & & & +=

Eg. Determine the magnetic flu0 density inside a closely wound toroidal coil with

an air core ha&ing N turns and carrying a current I . The toroid has a mean

radius b" and the radius of each turn is a.

('ol.) /) r&l o µ π ==⋅∫ 2dA

r

/) a &a &

o

π

µ φ φ

2 ==

, (ba)6r 6(bCa), 0= & for r 6(ba) and

r :(bCa)

Eg. 7n certain e0periments it is desira!le to ha&e a region of constant magnetic

flu0 density. This can !e created in an off-center cylindrical ca&ity. The uniform

a0ial current density is * z * = . Find the magnitude and direction of & in the

cylindrical ca&ity whose a0is is displaced from that of the conducting part !y a

distance d . 台大電研、清大電研、中原電機]

('ol.) * z * = , ) l & o µ =⋅∫ d

5f no hole eDists,

=

−=

⇒=⇒=

11

111

1

2

111

2

2

22

# *

&

y *

&

* r

& * r &r

o

y

o

#o

o

µ

µ

µ π µ π φ φ

7/23/2019 new EM1


For * in the hole potion,

−=

=

⇒−=22

22

2

2

2

2

2 #

* &

y *

&

*

r

&o

y

o

#o

µ

µ

µ

φ

+t 21 y y = and d # # += 21 021 =+=⇒ # # # & & & , and d

* & & & o y y y

221

µ =+=

7/23/2019 new EM1


1-8 Electromagnetic Forces

9orent force euation: )( &v E q F ×+=

Electric force: E q F e = . 4agnetic force: A×= vq F m

Eg. #n electron is in;ected with an initial &elocity

oo v yv =

into a region where!oth an electric field and a magnetic field & e0ist. Descri!e the motion of the

electron if o E z E = and o & # & = . Discuss the effect of the relati&e magnitude of

E ( and B( on the electron paths in parts.

('ol.) )( &v E et

vm ×+−=

∂∂

,

=−=

+−=

=

⇒

−−=

∂

∂

−=∂

∂

=∂

∂

⇒

=

=

oo

o

o

z

o

o

o

o

o y

#

yoo z

zo y

#

o

o

&

m

etv

R

Ev

&

Et

&

Evv

v

v & E

m

e

t

v

v &m

e

t

v

t

v

& # &

E z E

00

0

Esin)(

cos)(

0

)(

0

ω ω

ω

(5fo

oo &

E v ≠ )

22222

22

2 )()()(

),cos1(

sin

0

ooo

o

o

o

oo

o

o

o

o

o

cc zt

&

E y

&

E vct

c z

t &

E t

c y

#

ω ω

ω ω

ω ω

=++−⇒

−=−−=

+=

=

⇒

4agnetic force due to & and I :

& )d &d dt

dq &

dt

d dqdF &vq F mm

×=×=×=⇒×= , ∴ ∫ ×=$

m &d ) F

Eg. Determine the force per unit length !etween two infinitely long parallel

conducting wires carrying currents I 1 and I , in the same direction. The wires are

separated !y a distance d . $清大電研%

('ol.) )(' 12212 & z ) F ×= ,d ) ) y F

d ) # &

π µ

π µ

2'

2 210

1210

12 −=⇒−=

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Eg. Calculate the force per unit length on each of three euidistant" infinitely

long" parallel wires d apart" each carrying a current of I in the same direction.

pecify the direction of the force.

('ol.)

d

) y )& y & ) z -

d

) # & # & ) ) ) )

π

µ

π

µ

2

3,

2

330cos2,

2

0

222

0

122321 −=−=×−======

Eg. The !ar ##’" ser&es as a conducting path for the current I in two &ery long

parallel lines. The lines ha&e a radius b and are spaced at a distance d apart. Find

the direction and the magnitude of the magnetic force on the !ar. $中山物理所%

('ol.)

)

11

(4 0

yd y

)

z & −+−= π

µ

, dy yd =

)1ln(2

)11

(4

2

0

2

0 −−==⇒−

+−=×=⇒ ∫ −

b

d ) # F d F dy

yd y

) # & )d F d

bd

b π

µ

π

µ

#pplication of the electric forces: he epaper

7/23/2019 new EM1


calar electric potential function: ∫∫∫ ='

'4

1

"

dv R

" ρ

πε

http://faculty.pccu.edu.tw/~meng/e-paper.wmv

7/23/2019 new EM1


<ector magnetic potential function: ∫∫∫ ='

'4

"

dv R

* A

π

µ

=etarded potentials:

∫∫∫ −

='

')(

4

1),(

"

dv R

v Rt t R"

ρ

πε , ')/(4

),('

dv R

v Rt * t R A" ∫∫∫

−=

π µ

7/23/2019 new EM1


1-> Faraday’s 9aw and 4agnetic Dipoles

Faraday’s law: t

& E

∂∂

−=×∇

or ∫ ∫∫ ⋅∂∂

−=⋅$ S

S d t

&d E

∵ 0)()( =∂∂+×∇⇒×∇∂∂−=∂∂−=×∇t

A E At t

& E

∴ ∃" f!lfillst

A" E "

t

A E

∂

∂−−∇=⇒−∇=

∂

∂+

ote: 5n static field% " E −∇= , *!t in ti#e7ar"in$ field%t

A" E

∂∂

−−∇=

Emf : ∫ ⋅= l d E " . 4agnetic flu0: ∫∫ ⋅=ΦS

S d & , ∫ ∫∫ ⋅∂∂

−=⋅$ S

S d t

&d E

⇒

dt

d

"

Φ

−=

4otional emf : ∫ ⋅×=$

l d &v" )(' ("olt )

l d &vl d E " E &vq

F &vq F

$

mmm

m

⋅×=⋅−=⇒−≡×=⇒×= ∫ ∫ )('

Eg. # circular loop of N turns of conducting wire line in the xy-plane with its

center at the origin of a magnetic field specified !y t br & z & ω π sin)2cos(0= "

where b is the radius of the loop and ω is the angular freuency. Find the emf

induced in the loop.

('ol.) )sin()12

(G

2)sin()2

cos( 0

2

00 t &

brdr z t

b

r & z

b

ω π

π π ω

π −=⋅=Φ ∫

)cos()12

(G

0

2 t &b /

dt

/d " ω ω

π

π −−=

Φ−=

Eg. # metal !ar slides o&er a pair of conducting rails in a uniform magnetic field

0 & z & = with a constant &elocity . )a* Determine the open-circuit &oltage V ( that

appears across terminals 1 and ,. )!* #ssuming that a resistance R is connected!etween the two terminals" find the electric power dissipated in R. eglect the

electric resistance of the metal !ar and of the conducting rails. $交大電子物理

所%

('ol). (a)

∫ −=⋅×=−='1

'200210 )()( hv&dl y & z v #" " " (" )

(*)

R

hv& R

R

hv& R ) P e

2

0202 )()( === (+ )

7/23/2019 new EM1


Eg. The circuit in Fig. is situated in a magnetic field )3

210cos(3 #t z & π π −=

!" . #ssuming R'1>?" find the current # . $中山物理所%

('ol.) ∫ ⋅−×=Φ −

6.0

06 )2.0(10)

3210cos(3 d# #t π π

)10cos()6.03

210cos(4 9 t t

dt

d " π π π −−×=

Φ−= ,

)2.010sin(6.12

π π −== t R

" i

Eg. # conducting sliding !ar oscillates o&er two parallel conducting rails in a

sinusoidally &arying magnetic field )cos( t z & ω = " . The position of the sliding

!ar is gi&en !y x '(.5>)1-cosω$ *" and the rails are terminated in a resistance

R'(.,?. Find # .

('ol.) ).0(2.0cos #t −⋅=Φ ω , #=0.3(1cos0t ), dt

d

Ri

Φ−= 1

)cos21(sin.1 t t i ω ω ω +⋅=⇒

Eg. The Faraday dis3 generator consists of a circular metal dis3 rotating with a

constant angular &elocity @ in a uniform and constant magnetic field of flu0

density 0 & z & = that is parallel to the a0is of rotation. 6rush contacts are the

open-circuit &oltage of the generator if the radius of the dis3 is b.

('ol.) ∫ ∫ ∫ ==⋅×=⋅×=2

)()()(2

00

0

4

3 00

b &rdr &dr a & z r al d &v"

br

ω ω ω φ

(" )

7/23/2019 new EM1


4agnetic dipole moment: m ' m z )S z = , -here S is the area of the loop that carries

) and m= )S .

<ector potential of a magnetic dipole: 2

0

4

R

am A R

π

µ ×=

, -here

( )θ θ π

µ θ sincos2

4 3 aa

R

m A & R

o +=×∇=

Eg. For the small rectangular loop with sides a and b that carries a current I .

Find the &ector magnetic potential A at a distant point P ) x " y" z *. #nd determine

the magnetic flu0 density & and A . $交大光電所%

('ol.)

2

0

4

R

am

A

R

π

µ ×

=

, -here m= )ab, ( )θ θ π

µ θ sincos24 3 aa R

m

A & R

o

+=×∇=

4agnetiation &ector:

"

m

%

"

k

k

" ∆=

∑∆

=

→∆

1

0li#

( A/m), -here k m is the

#a$netic dipole #o#ent of an ato#.

×∇−×∇=∇×=

×=

R

% %

Rdv

R % dv

R

a % Ad

oo Ro ''1

4')

1('

4'

4

2

π

µ

π

µ

π

µ

∫∫∫ ∫∫∫

×∇−

×∇=⇒

' '

'4

''

4 " "

oo dv R

% dv

R

% A

π

µ

π

µ ( ''d'

'

dS F v F s"

×−=×∇

∫∫ ∫∫∫ )

∫∫ ∫∫∫ ×+

×∇= 'd

'

4'd

'

4'

S R

a % v

R

% no

"

o

π

µ

π

µ

∴ 9a$neti@ation 7ol!#e c!rrent densit"% % * m ×∇= ( A/m2)

9a$neti@ation s!rface c!rrent densit"% nms a % * ×= ( A/m)

Eui&alent 4agnetiation Charge Densities:

'd)'(

4

1'd

'

4

1

''

v

R

% S

R

a % "

" S

n

m ∫∫∫ ∫∫ ⋅∇

−+⋅

=

π π

(ote:

(∫∫∫ ∫∫

⋅∇−+

⋅=

'

''

4

1'

'

4

1

" o

n

o

dv R

P dS

R

a P "

πε πε )

Hefine the #a$neti@ation s!rface char$e densit" as nms a % ⋅= ρ and the

#a$neti@ation 7ol!#e char$e densit" as % m ⋅−∇= ρ

Eg. # circular rod of magnetic material with permea!ility ! is inserted coa0ially

in the long solenoid. The radius a of the rod is less than the inner radius b of the

solenoid. The solenoid’s winding has n turns per unit length and carries a

current I . )a* Find the &alues of &

" '

" and %

inside the solenoid for r Aa and

for aAr Ab. )!* Bhat are the eui&alent magnetiation current densities % m and % ms

7/23/2019 new EM1


for the magnetied rod2 $清大電研%

('ol.) (a) r 6 a% n) z ' = , n) z & µ = , n) z ' &

% )1(00

−=−= µ

µ

µ ,

a 6 r 6 b% n) z '

=, n) z &

o

µ = , 0

= %

(*) 0' =×∇= % * m , n) an) a z a % * r nms )1()1)((00

−=−×=×= µ

µ

µ

µ φ

Eg. # ferromagnetic sphere of radius b is uniformly magnetied with a

magnetiation 0 % z % = . )a* Determine the eui&alent magnetiation current

densities m * and ms * . )!* Determine the magnetic flu0 density at the center of

the sphere. $台大電研%

('ol.) (a) 0' =×∇= % * m , θ θ θ φ θ sin)sincos( o Ro Rms % aa % aa * =×−=

(*) θ µ θ θ µ 3

23

2

2

sin2

)(2

)sin)(d( oomso % zb

bb * z &d == , 000

3

32dsin

2 % z % z & oo µ θ θ µ π == ∫

.

Eg. Determine the magnetic flu0 density on the a0is of a uniformly magnetied

circular cylinder of a magnetic material. The cylinder has a radius b" length L

and a0ial magnetiation 0 % z % = . $台大物研%

('ol.) 0' =×∇= % * m " 00)( % aa % z a % * r nms φ =×=×= " 'dz * d) ms=

[ ]

+− −−+=+−= ∫ 222200

0 23

22

2

00

)(2

)'(2

'

b L z

L z

b z

z % z

b z z

dz b % z & L µ µ

7/23/2019 new EM1


Eg. # cylindrical !ar magnet of radius b and length L has a uniform

magnetiation 0 % z % = along its a0is. se the eui&alent magnetiation charge

density concept to determine the magnetic flu0 density at an ar!itrary distant

point. $交大電子所

%

('ol.)

−=⋅=

side.all

bottom %

to %

a % nms

,0

,

,

0

0

ρ " ρ#=0 in the interior re$ion

0

22 % bbq msm π ρ π == )11

(4

),,(−

−+

=⇒ R R

q z y #" m

mπ

( A)

2

2

2 4

cos)(

4

cos

R

L % b

R

Lq om

π

θ π

π

θ

== 24

cos

R

% 1

π

θ

= , -here 0

2

L% b % 1 π =

)sincos2(4 3

θ θ π

µ µ θ aa

R

% " & R

1 omo +=∇−= (1 )

>onsider an infinitel" lon$ solenoid -ith n t!rns per !nit len$th aro!nd to create a

#a$netic fieldE a 7olta$e " 1= dt d n Φ− is ind!ced !nit len$th, -hich opposes the

c!rrent chan$e. o-er P 1=" 1 ) per !nit len$th #!st *e s!pplied to o7erco#e this

ind!ced 7olta$e in order to increase the c!rrent to ) . he -orI per !nit 7ol!#e

re!ired to prod!ce a final #a$netic fl!D densit" &f is + 1= ∫ - &

'd&0

.

new em1

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